On Iwasawa \(\lambda^{-}\)-invariants of imaginary abelian fields
For a prime \(p\) and an imaginary abelian field \(k\), let \(\lambda_p(k)\) denote the Iwasawa \(\lambda\)-invariant of the basic \(\mathbb Z_p\)-extension \(k_{\infty}\) of \(k\). Put \(\lambda^-_p(k)=\lambda_p(k)- \lambda_p(k^+)\), where \(k^+\) denotes the maximal real subfield of \(k\). The author studies the distribution and range of values of \(\lambda^-_p(k)\) as \(k\) varies. He formulates the main results in three theorems, the first of which asserts the following: Among all cyclotomic fields \(k\) with conductor \(\leq x\), the proportion of fields having \(\lambda^-_p(k)\geq N\) (a fixed positive integer) tends to one as \(x\to \infty\). In fact, the result is proved in a somewhat more precise form. Let \(s_p(k)\) denote the number of primes of \((k^+)_{\infty}\) lying above \(p\) and decomposed in \(k_{\infty}\). Theorem 2: For any finite set \(P\) of prime numbers, there exists an infinite set \(QF\) of imaginary quadratic fields such that, given any positive integer \(n\), one has \(\lambda^-_v(K)\geq 2n\) and \(s_v(K)=0\) for every \(v\in P\) and for every compositum \(K\) of \(n\) distinct fields in \(QF\). Theorem 3: If \(p>2\) and \(n\) (\(\geq 0)\) is divisible by \(p-1\), then there exist infinitely many imaginary cyclic fields \(K\) of degree \(2p\) such that \(\lambda^-_p(K)=n\).
scientific article; zbMATH DE number 4019176
scientific article